8 XII TOPOLOGY AND TOPOLOGICAL ALGEBRA For E - U is closed in E, hence/(E - U) is closed in F by (12.3.6). We have b £/(E — U) by definition, hence the complement V of/(E — U) in F is an open neighborhood of b, and clearly U => /~1(V). PROBLEMS 1. Find all possible topologies on a set of 2 or 3 elements. 2. Let % be a set of subsets of a set E. Show that $ is the set of closed sets for a topology on E if and only if 3? satisfies the following two conditions: (1) every intersection of a family of sets belonging to $ belongs to $; (2) the empty set belongs to $, and the union of two sets belonging to % belongs to %. 3. Let E be a set and for each x e E let %$(x) be a set of subsets of E. Then there exists a topology & on E such that, for all x e E, %$(x) is the set of neighborhoods of x with respect to ^", if and only if the sets 85(#) satisfy the following conditions: (Vj) Every subset of E which contains a set belonging to %$(x) belongs to $(x). (VH) The intersection of two sets belonging to %$(x) belongs to %$(x). (Vm) For all x e E and all V e $(*), we have x e V. (VJV) For all x e E and all V e $(*), there exists a set W e %(x) such that V e %(y) for all y e W. The topology 9" is then unique. 4. Let A be a commutative ring with an identity element 1 =£ 0. An ideal $ of A is prime if A/£ is an integral domain (and therefore ^O). The set of prime ideals of A is called the spectrum of A and is denoted by Spec(A) (it can be shown to be nonempty). If a is an ideal in A, the set of prime ideals # => a is denoted by V(a). Show that V(a n 6) = V(a) u V(fc) for any two ideals a, fc in A. Deduce that the subsets V(a) of Spec(A) are the closed sets in a topology, called the spectral topology, on Spec(A). If x, y are two distinct points of Spec(A), show that either there is a neighborhood of x which does not contain y9 or else there is a neighborhood of y which does not contain x. Under what conditions is a set {x} consisting of a single point closed in Spec(A)? Consider the case where A = Z, and the case where A is a discrete valuation ring. Let A' be another ring and let h: A->A' be a ring homomorphism such that A(l)— 1. Show that the mapping £'(-» h~\%') of Spec(A') into Spec(A) is continuous with respect to the spectral topologies. 5. (a) The following conditions on a nonempty topological space E are equivalent: (1) the intersection of any two nonempty open sets in E is nonempty; (2) every non- empty open set is dense in E; (3) every open set in E is connected. The space E is then said to be irreducible. A nonempty subset F of a topological space E is said to be an irreducible set if the subspace F is irreducible. (b) Show that in a Hausdorff space every irreducible set consists of a single point.